Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 7H (Statistics)

For observations $x_1,\dots ,x_n$, the log-likelihood of the normal distribution model isDifferentiating first with respect to $\mu$ gives $\hat{\mu }=\overline{x}$. Substituting this value and differentiating with respect to ${\sigma }^2$ gives the normal mean and variance maximum-likelihood estimators

An unbiased estimator $T$ of a parameter $\theta$ satisfies ${\mathbb{E}}_{\theta }[T]=\theta$ for every permissible value of $\theta$. Since the sample mean is linear,so $\hat{\mu }$ is unbiased.

Use the identityThe two expectations on the right are $n{\sigma }^2$ and $n\mathrm{Var}(\overline{X})={\sigma }^2$, respectively. HenceThusThe unbiased sample variance instead divides the same sum of squares by $n-1$.